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Consider the following. Differential Equation Solutions y"" + 6y" +9y' = 0 {e-3x, xe-3x, (3x + 1)e-³x) (a) Verify that each solution satisfies the differential equation. y = e-3x

Consider the following. Differential Equation Solutions y"" + 6y" +9y' = 0 {e-3x, xe-3x, (3x + 1)e-³x) (a) Verify that each solution satisfies the differential equation. y = e-3x

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Certainly! Below is the transcription suitable for an educational website: --- **Consider the following.** **Differential Equation** \[ y''' + 6y'' + 9y' = 0 \] **Solutions** \[ \{e^{-3x}, xe^{-3x}, (3x + 1)e^{-3x}\} \] **(a) Verify that each solution satisfies the differential equation.** 1. \( y = e^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification] 2. \( y = xe^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification] 3. \( y = (3x + 1)e^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification] --- This exercise involves verifying the given solutions by computing their derivatives and substituting them back into the original differential equation to ensure they satisfy it.
Transcribed Image Text:Certainly! Below is the transcription suitable for an educational website: --- **Consider the following.** **Differential Equation** \[ y''' + 6y'' + 9y' = 0 \] **Solutions** \[ \{e^{-3x}, xe^{-3x}, (3x + 1)e^{-3x}\} \] **(a) Verify that each solution satisfies the differential equation.** 1. \( y = e^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification] 2. \( y = xe^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification] 3. \( y = (3x + 1)e^{-3x} \) - \( y' = \) [Box for entering derivative] - \( y'' = \) [Box for entering second derivative] - \( y''' = \) [Box for entering third derivative] - \( y''' + 6y'' + 9y' = \) [Box for showing verification] --- This exercise involves verifying the given solutions by computing their derivatives and substituting them back into the original differential equation to ensure they satisfy it.
The image displays an exercise related to differential equations. The task involves solving the following problems: 1. **Differential Equation:** - \( y''' + 6y'' + 9y' = \) [Input Box] 2. **Linear Independence Test:** - The user must test the set of solutions for linear independence by choosing one of the two options: - ( ) linearly independent - ( ) linearly dependent 3. **General Solution:** - If the set is deemed linearly independent, users are to input the general solution of the differential equation. Instructions note that if the system is dependent, the user should enter "DEPENDENT." Constants \( C_1 \) and \( C_2 \) may be used as needed in the general solution: - \( y = \) [Input Box] 4. **Submission Button:** - "Submit Answer" is mentioned at the bottom for submitting responses. There are no graphs or diagrams present in the image.
Transcribed Image Text:The image displays an exercise related to differential equations. The task involves solving the following problems: 1. **Differential Equation:** - \( y''' + 6y'' + 9y' = \) [Input Box] 2. **Linear Independence Test:** - The user must test the set of solutions for linear independence by choosing one of the two options: - ( ) linearly independent - ( ) linearly dependent 3. **General Solution:** - If the set is deemed linearly independent, users are to input the general solution of the differential equation. Instructions note that if the system is dependent, the user should enter "DEPENDENT." Constants \( C_1 \) and \( C_2 \) may be used as needed in the general solution: - \( y = \) [Input Box] 4. **Submission Button:** - "Submit Answer" is mentioned at the bottom for submitting responses. There are no graphs or diagrams present in the image.
Expert Solution
Step 1: Introduction of the given problem

y apostrophe apostrophe apostrophe plus 6 y apostrophe apostrophe plus 9 y apostrophe equals 0

The set of solution is open curly brackets e to the power of negative 3 x end exponent comma space x e to the power of negative 3 x end exponent comma space open parentheses 3 x plus 1 close parentheses e to the power of negative 3 x end exponent close curly brackets

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